3.3 Free field realizations
3.3.2 Spectral flow in the free field realization
The bosonic realization (3.165) of the SL(2,R) current algebra provides an interesting tool to examine strings on the BTZ black hole. Since we have seen that the spectral flow is an integral feature in the construction of the string spectrum on the SL(2,R) group manifold, there is a need to clarify the role of spectral flow in the free field realization.
We approach this problem by considering the global properties of the BTZ geometry and solving the periodic identifications for the fields Xa [57]. In this way, we obtain the same values for the spectral flow parameters as in section 3.2.2.
First, we observe that the sigma model form of the WZW action (3.139) can be converted to the BTZ metric (2.24) using the following transformation:
ψ =
The periodic identification of the angular BTZ coordinateθ ∼θ+ 2π translates into the following identifications of (φ, γ,¯γ):
(γ(z),γ(¯¯ z), φ(z,z))¯ ∼¡
γ(z)e2π∆− , ¯γ(¯z)e2π∆+ , φ(z,z)¯ −2πr+
¢ (3.169)
Notice that in the free field limit, γ = ψ(z) is a function of z only, and ¯γ = χ(¯z) is a function of ¯z only, respectively.
In order to find out the periodicities of the fields Xa, we perform the following trans-formations:
With these transformations, we arrive at the currents (3.165). Bearing in mind that we have rescaled the fieldφ →φ/√
2k0 to obtain the current algebra (3.147), the identification of φ also gets rescaled:
(γ(z),¯γ(¯z), φ(z,z))¯ ∼ We find that the periodicity condition (3.171) is satisfied if6
³ Notice that the factors includingr−cancel in the identification ofφ(z,z) since we have to¯ include both the holomorphic and the antiholomorphic part ofX1(z,z) in the calculation.¯
6It might appear that the periodicity condition could be imposed on the lightcone coordinateX−: (X−(z),X˜−(¯z))∼(X−(z) +π√
2k∆− , X˜−(¯z) +π√
2k ∆+). However, this would include an unwanted term proportional to −ir+
√2klnz in the mode expansion ofX0, that conflicts with the periodicity of the fieldβ (3.170).
The result (3.172) tells us that the field X1 corresponds to a compact coordinate. A natural interpretation is that the U(1) current J2 generated by ∂X1 describes rotations in the compact θ-direction of the BTZ black hole geometry. We need to incorporate the effect of compactness in the mode expansion of X1. This is done by including a shift in the zero modes of the fieldX1:
X1(z,z) = ˆ¯ q1− i 2
³ ˆ
p1−w+√ 2k
´
lnz− i 2
³ˆ¯p1+w−√ 2k
´
ln¯z+ oscillations (3.173) We have suggestively labeledw± =m∆∓ (m ∈Z). The apparent difference between the radii of the holomorphic and the antiholomorphic part is an effect of the intrinsic angular momentum of the target space. For a non-rotating black hole, r− = 0, and the radii become equal.
What is the meaning of the transformation
³ ˆ p1,ˆ¯p1
´
→
³ ˆ
p1−w+√
2k , ˆ¯p1+w−√ 2k
´
? Let us see how this affects the currents (3.165). We find the following transformation rules for the modes of the currents:
Jn± → Jn±iw± + , J¯n± → J¯n±iw± −
Jn2 → Jn2+k2w+δn,0 , J¯n2 → J¯n2− k2w−δn,0 (3.174) These look familiar. In fact, we have just reproduced the transformation law (3.99) of spectral flow for the modes of the currents. Hence, the spectral flow is generated by twisting of the field X1 in the free field realization (3.165).
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